Properties

Label 1380.2.t.a.1333.23
Level $1380$
Weight $2$
Character 1380.1333
Analytic conductor $11.019$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1380,2,Mod(1057,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.1057");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.t (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1333.23
Character \(\chi\) \(=\) 1380.1333
Dual form 1380.2.t.a.1057.23

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 + 0.707107i) q^{3} +(-1.72321 + 1.42497i) q^{5} +(-1.66865 - 1.66865i) q^{7} +1.00000i q^{9} +O(q^{10})\) \(q+(0.707107 + 0.707107i) q^{3} +(-1.72321 + 1.42497i) q^{5} +(-1.66865 - 1.66865i) q^{7} +1.00000i q^{9} -0.0251310i q^{11} +(4.31706 + 4.31706i) q^{13} +(-2.22610 - 0.210891i) q^{15} +(2.93368 + 2.93368i) q^{17} -5.73401 q^{19} -2.35982i q^{21} +(-1.66047 - 4.49920i) q^{23} +(0.938928 - 4.91105i) q^{25} +(-0.707107 + 0.707107i) q^{27} +7.21783i q^{29} -8.64554 q^{31} +(0.0177703 - 0.0177703i) q^{33} +(5.25321 + 0.497665i) q^{35} +(-6.47729 - 6.47729i) q^{37} +6.10524i q^{39} -0.0620681 q^{41} +(-5.43676 + 5.43676i) q^{43} +(-1.42497 - 1.72321i) q^{45} +(-4.36786 + 4.36786i) q^{47} -1.43123i q^{49} +4.14884i q^{51} +(-9.27165 + 9.27165i) q^{53} +(0.0358109 + 0.0433061i) q^{55} +(-4.05455 - 4.05455i) q^{57} +2.10077i q^{59} +4.81366i q^{61} +(1.66865 - 1.66865i) q^{63} +(-13.5909 - 1.28754i) q^{65} +(0.147877 + 0.147877i) q^{67} +(2.00729 - 4.35555i) q^{69} -4.45924 q^{71} +(-1.58272 - 1.58272i) q^{73} +(4.13656 - 2.80871i) q^{75} +(-0.0419348 + 0.0419348i) q^{77} +5.73841 q^{79} -1.00000 q^{81} +(0.182671 - 0.182671i) q^{83} +(-9.23574 - 0.874953i) q^{85} +(-5.10377 + 5.10377i) q^{87} +10.2895 q^{89} -14.4073i q^{91} +(-6.11332 - 6.11332i) q^{93} +(9.88091 - 8.17078i) q^{95} +(-12.8876 - 12.8876i) q^{97} +0.0251310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 8 q^{13} - 16 q^{23} - 8 q^{25} + 8 q^{31} + 8 q^{35} - 24 q^{41} + 8 q^{47} - 32 q^{55} - 24 q^{71} + 8 q^{73} + 32 q^{75} + 40 q^{77} - 48 q^{81} + 24 q^{85} - 40 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.707107 + 0.707107i 0.408248 + 0.408248i
\(4\) 0 0
\(5\) −1.72321 + 1.42497i −0.770644 + 0.637265i
\(6\) 0 0
\(7\) −1.66865 1.66865i −0.630690 0.630690i 0.317552 0.948241i \(-0.397139\pi\)
−0.948241 + 0.317552i \(0.897139\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 0.0251310i 0.00757729i −0.999993 0.00378865i \(-0.998794\pi\)
0.999993 0.00378865i \(-0.00120597\pi\)
\(12\) 0 0
\(13\) 4.31706 + 4.31706i 1.19734 + 1.19734i 0.974961 + 0.222375i \(0.0713810\pi\)
0.222375 + 0.974961i \(0.428619\pi\)
\(14\) 0 0
\(15\) −2.22610 0.210891i −0.574777 0.0544518i
\(16\) 0 0
\(17\) 2.93368 + 2.93368i 0.711521 + 0.711521i 0.966853 0.255333i \(-0.0821849\pi\)
−0.255333 + 0.966853i \(0.582185\pi\)
\(18\) 0 0
\(19\) −5.73401 −1.31547 −0.657736 0.753249i \(-0.728486\pi\)
−0.657736 + 0.753249i \(0.728486\pi\)
\(20\) 0 0
\(21\) 2.35982i 0.514956i
\(22\) 0 0
\(23\) −1.66047 4.49920i −0.346232 0.938149i
\(24\) 0 0
\(25\) 0.938928 4.91105i 0.187786 0.982210i
\(26\) 0 0
\(27\) −0.707107 + 0.707107i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 7.21783i 1.34032i 0.742218 + 0.670158i \(0.233774\pi\)
−0.742218 + 0.670158i \(0.766226\pi\)
\(30\) 0 0
\(31\) −8.64554 −1.55278 −0.776392 0.630250i \(-0.782952\pi\)
−0.776392 + 0.630250i \(0.782952\pi\)
\(32\) 0 0
\(33\) 0.0177703 0.0177703i 0.00309342 0.00309342i
\(34\) 0 0
\(35\) 5.25321 + 0.497665i 0.887954 + 0.0841208i
\(36\) 0 0
\(37\) −6.47729 6.47729i −1.06486 1.06486i −0.997745 0.0671152i \(-0.978620\pi\)
−0.0671152 0.997745i \(-0.521380\pi\)
\(38\) 0 0
\(39\) 6.10524i 0.977621i
\(40\) 0 0
\(41\) −0.0620681 −0.00969341 −0.00484671 0.999988i \(-0.501543\pi\)
−0.00484671 + 0.999988i \(0.501543\pi\)
\(42\) 0 0
\(43\) −5.43676 + 5.43676i −0.829097 + 0.829097i −0.987392 0.158295i \(-0.949400\pi\)
0.158295 + 0.987392i \(0.449400\pi\)
\(44\) 0 0
\(45\) −1.42497 1.72321i −0.212422 0.256881i
\(46\) 0 0
\(47\) −4.36786 + 4.36786i −0.637117 + 0.637117i −0.949843 0.312726i \(-0.898758\pi\)
0.312726 + 0.949843i \(0.398758\pi\)
\(48\) 0 0
\(49\) 1.43123i 0.204461i
\(50\) 0 0
\(51\) 4.14884i 0.580954i
\(52\) 0 0
\(53\) −9.27165 + 9.27165i −1.27356 + 1.27356i −0.329352 + 0.944207i \(0.606830\pi\)
−0.944207 + 0.329352i \(0.893170\pi\)
\(54\) 0 0
\(55\) 0.0358109 + 0.0433061i 0.00482875 + 0.00583940i
\(56\) 0 0
\(57\) −4.05455 4.05455i −0.537039 0.537039i
\(58\) 0 0
\(59\) 2.10077i 0.273497i 0.990606 + 0.136748i \(0.0436652\pi\)
−0.990606 + 0.136748i \(0.956335\pi\)
\(60\) 0 0
\(61\) 4.81366i 0.616327i 0.951333 + 0.308163i \(0.0997143\pi\)
−0.951333 + 0.308163i \(0.900286\pi\)
\(62\) 0 0
\(63\) 1.66865 1.66865i 0.210230 0.210230i
\(64\) 0 0
\(65\) −13.5909 1.28754i −1.68574 0.159700i
\(66\) 0 0
\(67\) 0.147877 + 0.147877i 0.0180661 + 0.0180661i 0.716082 0.698016i \(-0.245934\pi\)
−0.698016 + 0.716082i \(0.745934\pi\)
\(68\) 0 0
\(69\) 2.00729 4.35555i 0.241649 0.524346i
\(70\) 0 0
\(71\) −4.45924 −0.529215 −0.264607 0.964356i \(-0.585242\pi\)
−0.264607 + 0.964356i \(0.585242\pi\)
\(72\) 0 0
\(73\) −1.58272 1.58272i −0.185243 0.185243i 0.608393 0.793636i \(-0.291814\pi\)
−0.793636 + 0.608393i \(0.791814\pi\)
\(74\) 0 0
\(75\) 4.13656 2.80871i 0.477649 0.324322i
\(76\) 0 0
\(77\) −0.0419348 + 0.0419348i −0.00477892 + 0.00477892i
\(78\) 0 0
\(79\) 5.73841 0.645621 0.322811 0.946464i \(-0.395372\pi\)
0.322811 + 0.946464i \(0.395372\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 0.182671 0.182671i 0.0200507 0.0200507i −0.697010 0.717061i \(-0.745487\pi\)
0.717061 + 0.697010i \(0.245487\pi\)
\(84\) 0 0
\(85\) −9.23574 0.874953i −1.00176 0.0949019i
\(86\) 0 0
\(87\) −5.10377 + 5.10377i −0.547182 + 0.547182i
\(88\) 0 0
\(89\) 10.2895 1.09069 0.545343 0.838213i \(-0.316399\pi\)
0.545343 + 0.838213i \(0.316399\pi\)
\(90\) 0 0
\(91\) 14.4073i 1.51029i
\(92\) 0 0
\(93\) −6.11332 6.11332i −0.633922 0.633922i
\(94\) 0 0
\(95\) 9.88091 8.17078i 1.01376 0.838304i
\(96\) 0 0
\(97\) −12.8876 12.8876i −1.30854 1.30854i −0.922470 0.386069i \(-0.873833\pi\)
−0.386069 0.922470i \(-0.626167\pi\)
\(98\) 0 0
\(99\) 0.0251310 0.00252576
\(100\) 0 0
\(101\) 15.7220 1.56440 0.782201 0.623026i \(-0.214097\pi\)
0.782201 + 0.623026i \(0.214097\pi\)
\(102\) 0 0
\(103\) −2.79500 + 2.79500i −0.275399 + 0.275399i −0.831269 0.555870i \(-0.812385\pi\)
0.555870 + 0.831269i \(0.312385\pi\)
\(104\) 0 0
\(105\) 3.36268 + 4.06648i 0.328164 + 0.396848i
\(106\) 0 0
\(107\) 0.00287023 + 0.00287023i 0.000277476 + 0.000277476i 0.707246 0.706968i \(-0.249938\pi\)
−0.706968 + 0.707246i \(0.749938\pi\)
\(108\) 0 0
\(109\) −4.83662 −0.463264 −0.231632 0.972803i \(-0.574407\pi\)
−0.231632 + 0.972803i \(0.574407\pi\)
\(110\) 0 0
\(111\) 9.16028i 0.869455i
\(112\) 0 0
\(113\) 3.78998 3.78998i 0.356531 0.356531i −0.506002 0.862533i \(-0.668877\pi\)
0.862533 + 0.506002i \(0.168877\pi\)
\(114\) 0 0
\(115\) 9.27257 + 5.38697i 0.864671 + 0.502338i
\(116\) 0 0
\(117\) −4.31706 + 4.31706i −0.399112 + 0.399112i
\(118\) 0 0
\(119\) 9.79054i 0.897497i
\(120\) 0 0
\(121\) 10.9994 0.999943
\(122\) 0 0
\(123\) −0.0438888 0.0438888i −0.00395732 0.00395732i
\(124\) 0 0
\(125\) 5.38012 + 9.80073i 0.481213 + 0.876604i
\(126\) 0 0
\(127\) −5.03114 + 5.03114i −0.446442 + 0.446442i −0.894170 0.447728i \(-0.852233\pi\)
0.447728 + 0.894170i \(0.352233\pi\)
\(128\) 0 0
\(129\) −7.68873 −0.676955
\(130\) 0 0
\(131\) −1.17258 −0.102449 −0.0512245 0.998687i \(-0.516312\pi\)
−0.0512245 + 0.998687i \(0.516312\pi\)
\(132\) 0 0
\(133\) 9.56803 + 9.56803i 0.829654 + 0.829654i
\(134\) 0 0
\(135\) 0.210891 2.22610i 0.0181506 0.191592i
\(136\) 0 0
\(137\) 16.1030 + 16.1030i 1.37578 + 1.37578i 0.851627 + 0.524149i \(0.175616\pi\)
0.524149 + 0.851627i \(0.324384\pi\)
\(138\) 0 0
\(139\) 18.7001i 1.58612i 0.609145 + 0.793059i \(0.291513\pi\)
−0.609145 + 0.793059i \(0.708487\pi\)
\(140\) 0 0
\(141\) −6.17708 −0.520204
\(142\) 0 0
\(143\) 0.108492 0.108492i 0.00907257 0.00907257i
\(144\) 0 0
\(145\) −10.2852 12.4379i −0.854137 1.03291i
\(146\) 0 0
\(147\) 1.01203 1.01203i 0.0834710 0.0834710i
\(148\) 0 0
\(149\) −3.92087 −0.321210 −0.160605 0.987019i \(-0.551344\pi\)
−0.160605 + 0.987019i \(0.551344\pi\)
\(150\) 0 0
\(151\) 6.38226 0.519381 0.259691 0.965692i \(-0.416379\pi\)
0.259691 + 0.965692i \(0.416379\pi\)
\(152\) 0 0
\(153\) −2.93368 + 2.93368i −0.237174 + 0.237174i
\(154\) 0 0
\(155\) 14.8981 12.3196i 1.19664 0.989536i
\(156\) 0 0
\(157\) 10.0603 + 10.0603i 0.802901 + 0.802901i 0.983548 0.180647i \(-0.0578191\pi\)
−0.180647 + 0.983548i \(0.557819\pi\)
\(158\) 0 0
\(159\) −13.1121 −1.03986
\(160\) 0 0
\(161\) −4.73685 + 10.2783i −0.373316 + 0.810045i
\(162\) 0 0
\(163\) −15.5575 15.5575i −1.21856 1.21856i −0.968137 0.250423i \(-0.919430\pi\)
−0.250423 0.968137i \(-0.580570\pi\)
\(164\) 0 0
\(165\) −0.00529990 + 0.0559442i −0.000412597 + 0.00435525i
\(166\) 0 0
\(167\) 12.5827 12.5827i 0.973682 0.973682i −0.0259809 0.999662i \(-0.508271\pi\)
0.999662 + 0.0259809i \(0.00827092\pi\)
\(168\) 0 0
\(169\) 24.2740i 1.86723i
\(170\) 0 0
\(171\) 5.73401i 0.438490i
\(172\) 0 0
\(173\) −5.47575 5.47575i −0.416314 0.416314i 0.467617 0.883931i \(-0.345113\pi\)
−0.883931 + 0.467617i \(0.845113\pi\)
\(174\) 0 0
\(175\) −9.76155 + 6.62807i −0.737904 + 0.501035i
\(176\) 0 0
\(177\) −1.48547 + 1.48547i −0.111655 + 0.111655i
\(178\) 0 0
\(179\) 5.39989i 0.403607i 0.979426 + 0.201804i \(0.0646802\pi\)
−0.979426 + 0.201804i \(0.935320\pi\)
\(180\) 0 0
\(181\) 3.85131i 0.286266i −0.989703 0.143133i \(-0.954282\pi\)
0.989703 0.143133i \(-0.0457176\pi\)
\(182\) 0 0
\(183\) −3.40377 + 3.40377i −0.251614 + 0.251614i
\(184\) 0 0
\(185\) 20.3917 + 1.93182i 1.49923 + 0.142030i
\(186\) 0 0
\(187\) 0.0737263 0.0737263i 0.00539140 0.00539140i
\(188\) 0 0
\(189\) 2.35982 0.171652
\(190\) 0 0
\(191\) 2.52181i 0.182472i 0.995829 + 0.0912360i \(0.0290818\pi\)
−0.995829 + 0.0912360i \(0.970918\pi\)
\(192\) 0 0
\(193\) −16.4431 16.4431i −1.18360 1.18360i −0.978804 0.204799i \(-0.934346\pi\)
−0.204799 0.978804i \(-0.565654\pi\)
\(194\) 0 0
\(195\) −8.69978 10.5206i −0.623004 0.753398i
\(196\) 0 0
\(197\) −11.0590 + 11.0590i −0.787924 + 0.787924i −0.981154 0.193230i \(-0.938104\pi\)
0.193230 + 0.981154i \(0.438104\pi\)
\(198\) 0 0
\(199\) 12.5439 0.889216 0.444608 0.895725i \(-0.353343\pi\)
0.444608 + 0.895725i \(0.353343\pi\)
\(200\) 0 0
\(201\) 0.209130i 0.0147509i
\(202\) 0 0
\(203\) 12.0440 12.0440i 0.845324 0.845324i
\(204\) 0 0
\(205\) 0.106957 0.0884451i 0.00747017 0.00617728i
\(206\) 0 0
\(207\) 4.49920 1.66047i 0.312716 0.115411i
\(208\) 0 0
\(209\) 0.144101i 0.00996771i
\(210\) 0 0
\(211\) 8.92577 0.614475 0.307238 0.951633i \(-0.400595\pi\)
0.307238 + 0.951633i \(0.400595\pi\)
\(212\) 0 0
\(213\) −3.15316 3.15316i −0.216051 0.216051i
\(214\) 0 0
\(215\) 1.62148 17.1159i 0.110584 1.16729i
\(216\) 0 0
\(217\) 14.4264 + 14.4264i 0.979325 + 0.979325i
\(218\) 0 0
\(219\) 2.23830i 0.151250i
\(220\) 0 0
\(221\) 25.3297i 1.70386i
\(222\) 0 0
\(223\) −8.09635 8.09635i −0.542172 0.542172i 0.381993 0.924165i \(-0.375238\pi\)
−0.924165 + 0.381993i \(0.875238\pi\)
\(224\) 0 0
\(225\) 4.91105 + 0.938928i 0.327403 + 0.0625952i
\(226\) 0 0
\(227\) 19.1918 + 19.1918i 1.27380 + 1.27380i 0.944077 + 0.329724i \(0.106956\pi\)
0.329724 + 0.944077i \(0.393044\pi\)
\(228\) 0 0
\(229\) 17.7187 1.17088 0.585442 0.810714i \(-0.300921\pi\)
0.585442 + 0.810714i \(0.300921\pi\)
\(230\) 0 0
\(231\) −0.0593048 −0.00390197
\(232\) 0 0
\(233\) −2.12902 2.12902i −0.139477 0.139477i 0.633921 0.773398i \(-0.281444\pi\)
−0.773398 + 0.633921i \(0.781444\pi\)
\(234\) 0 0
\(235\) 1.30269 13.7508i 0.0849781 0.897004i
\(236\) 0 0
\(237\) 4.05767 + 4.05767i 0.263574 + 0.263574i
\(238\) 0 0
\(239\) 24.5239i 1.58632i 0.609013 + 0.793161i \(0.291566\pi\)
−0.609013 + 0.793161i \(0.708434\pi\)
\(240\) 0 0
\(241\) 18.6804i 1.20331i 0.798757 + 0.601654i \(0.205491\pi\)
−0.798757 + 0.601654i \(0.794509\pi\)
\(242\) 0 0
\(243\) −0.707107 0.707107i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) 2.03946 + 2.46632i 0.130296 + 0.157567i
\(246\) 0 0
\(247\) −24.7540 24.7540i −1.57506 1.57506i
\(248\) 0 0
\(249\) 0.258336 0.0163714
\(250\) 0 0
\(251\) 2.61887i 0.165302i −0.996579 0.0826509i \(-0.973661\pi\)
0.996579 0.0826509i \(-0.0263386\pi\)
\(252\) 0 0
\(253\) −0.113070 + 0.0417293i −0.00710863 + 0.00262350i
\(254\) 0 0
\(255\) −5.91197 7.14934i −0.370222 0.447709i
\(256\) 0 0
\(257\) 20.9282 20.9282i 1.30546 1.30546i 0.380811 0.924653i \(-0.375645\pi\)
0.924653 0.380811i \(-0.124355\pi\)
\(258\) 0 0
\(259\) 21.6166i 1.34319i
\(260\) 0 0
\(261\) −7.21783 −0.446772
\(262\) 0 0
\(263\) −5.24246 + 5.24246i −0.323264 + 0.323264i −0.850018 0.526754i \(-0.823409\pi\)
0.526754 + 0.850018i \(0.323409\pi\)
\(264\) 0 0
\(265\) 2.76522 29.1888i 0.169866 1.79306i
\(266\) 0 0
\(267\) 7.27579 + 7.27579i 0.445271 + 0.445271i
\(268\) 0 0
\(269\) 16.4697i 1.00418i −0.864816 0.502088i \(-0.832565\pi\)
0.864816 0.502088i \(-0.167435\pi\)
\(270\) 0 0
\(271\) −16.7600 −1.01810 −0.509049 0.860738i \(-0.670003\pi\)
−0.509049 + 0.860738i \(0.670003\pi\)
\(272\) 0 0
\(273\) 10.1875 10.1875i 0.616575 0.616575i
\(274\) 0 0
\(275\) −0.123420 0.0235962i −0.00744249 0.00142291i
\(276\) 0 0
\(277\) −14.8812 + 14.8812i −0.894125 + 0.894125i −0.994908 0.100784i \(-0.967865\pi\)
0.100784 + 0.994908i \(0.467865\pi\)
\(278\) 0 0
\(279\) 8.64554i 0.517595i
\(280\) 0 0
\(281\) 3.06746i 0.182989i 0.995806 + 0.0914946i \(0.0291644\pi\)
−0.995806 + 0.0914946i \(0.970836\pi\)
\(282\) 0 0
\(283\) 6.60619 6.60619i 0.392697 0.392697i −0.482951 0.875648i \(-0.660435\pi\)
0.875648 + 0.482951i \(0.160435\pi\)
\(284\) 0 0
\(285\) 12.7645 + 1.20925i 0.756102 + 0.0716297i
\(286\) 0 0
\(287\) 0.103570 + 0.103570i 0.00611353 + 0.00611353i
\(288\) 0 0
\(289\) 0.212901i 0.0125236i
\(290\) 0 0
\(291\) 18.2258i 1.06842i
\(292\) 0 0
\(293\) −21.3967 + 21.3967i −1.25001 + 1.25001i −0.294295 + 0.955715i \(0.595085\pi\)
−0.955715 + 0.294295i \(0.904915\pi\)
\(294\) 0 0
\(295\) −2.99353 3.62007i −0.174290 0.210769i
\(296\) 0 0
\(297\) 0.0177703 + 0.0177703i 0.00103114 + 0.00103114i
\(298\) 0 0
\(299\) 12.2550 26.5917i 0.708724 1.53784i
\(300\) 0 0
\(301\) 18.1441 1.04581
\(302\) 0 0
\(303\) 11.1172 + 11.1172i 0.638664 + 0.638664i
\(304\) 0 0
\(305\) −6.85932 8.29497i −0.392764 0.474969i
\(306\) 0 0
\(307\) 9.43070 9.43070i 0.538238 0.538238i −0.384773 0.923011i \(-0.625720\pi\)
0.923011 + 0.384773i \(0.125720\pi\)
\(308\) 0 0
\(309\) −3.95273 −0.224863
\(310\) 0 0
\(311\) −18.0668 −1.02448 −0.512238 0.858844i \(-0.671183\pi\)
−0.512238 + 0.858844i \(0.671183\pi\)
\(312\) 0 0
\(313\) 1.12544 1.12544i 0.0636139 0.0636139i −0.674584 0.738198i \(-0.735677\pi\)
0.738198 + 0.674584i \(0.235677\pi\)
\(314\) 0 0
\(315\) −0.497665 + 5.25321i −0.0280403 + 0.295985i
\(316\) 0 0
\(317\) −8.91087 + 8.91087i −0.500484 + 0.500484i −0.911588 0.411104i \(-0.865143\pi\)
0.411104 + 0.911588i \(0.365143\pi\)
\(318\) 0 0
\(319\) 0.181391 0.0101560
\(320\) 0 0
\(321\) 0.00405912i 0.000226558i
\(322\) 0 0
\(323\) −16.8217 16.8217i −0.935985 0.935985i
\(324\) 0 0
\(325\) 25.2547 17.1479i 1.40088 0.951193i
\(326\) 0 0
\(327\) −3.42001 3.42001i −0.189127 0.189127i
\(328\) 0 0
\(329\) 14.5768 0.803646
\(330\) 0 0
\(331\) 18.0284 0.990929 0.495465 0.868628i \(-0.334998\pi\)
0.495465 + 0.868628i \(0.334998\pi\)
\(332\) 0 0
\(333\) 6.47729 6.47729i 0.354953 0.354953i
\(334\) 0 0
\(335\) −0.465545 0.0441037i −0.0254355 0.00240964i
\(336\) 0 0
\(337\) 22.5933 + 22.5933i 1.23074 + 1.23074i 0.963681 + 0.267055i \(0.0860505\pi\)
0.267055 + 0.963681i \(0.413950\pi\)
\(338\) 0 0
\(339\) 5.35984 0.291106
\(340\) 0 0
\(341\) 0.217271i 0.0117659i
\(342\) 0 0
\(343\) −14.0688 + 14.0688i −0.759641 + 0.759641i
\(344\) 0 0
\(345\) 2.74753 + 10.3659i 0.147922 + 0.558079i
\(346\) 0 0
\(347\) −3.59353 + 3.59353i −0.192911 + 0.192911i −0.796953 0.604042i \(-0.793556\pi\)
0.604042 + 0.796953i \(0.293556\pi\)
\(348\) 0 0
\(349\) 0.339995i 0.0181995i 0.999959 + 0.00909975i \(0.00289658\pi\)
−0.999959 + 0.00909975i \(0.997103\pi\)
\(350\) 0 0
\(351\) −6.10524 −0.325874
\(352\) 0 0
\(353\) 15.6225 + 15.6225i 0.831499 + 0.831499i 0.987722 0.156223i \(-0.0499317\pi\)
−0.156223 + 0.987722i \(0.549932\pi\)
\(354\) 0 0
\(355\) 7.68423 6.35428i 0.407837 0.337250i
\(356\) 0 0
\(357\) 6.92296 6.92296i 0.366402 0.366402i
\(358\) 0 0
\(359\) 20.3053 1.07168 0.535838 0.844321i \(-0.319996\pi\)
0.535838 + 0.844321i \(0.319996\pi\)
\(360\) 0 0
\(361\) 13.8788 0.730464
\(362\) 0 0
\(363\) 7.77773 + 7.77773i 0.408225 + 0.408225i
\(364\) 0 0
\(365\) 4.98268 + 0.472037i 0.260806 + 0.0247075i
\(366\) 0 0
\(367\) −11.0788 11.0788i −0.578307 0.578307i 0.356130 0.934436i \(-0.384096\pi\)
−0.934436 + 0.356130i \(0.884096\pi\)
\(368\) 0 0
\(369\) 0.0620681i 0.00323114i
\(370\) 0 0
\(371\) 30.9422 1.60644
\(372\) 0 0
\(373\) 4.21203 4.21203i 0.218091 0.218091i −0.589603 0.807693i \(-0.700716\pi\)
0.807693 + 0.589603i \(0.200716\pi\)
\(374\) 0 0
\(375\) −3.12584 + 10.7345i −0.161418 + 0.554326i
\(376\) 0 0
\(377\) −31.1598 + 31.1598i −1.60481 + 1.60481i
\(378\) 0 0
\(379\) 38.3993 1.97244 0.986221 0.165435i \(-0.0529029\pi\)
0.986221 + 0.165435i \(0.0529029\pi\)
\(380\) 0 0
\(381\) −7.11511 −0.364518
\(382\) 0 0
\(383\) 7.00924 7.00924i 0.358155 0.358155i −0.504977 0.863133i \(-0.668499\pi\)
0.863133 + 0.504977i \(0.168499\pi\)
\(384\) 0 0
\(385\) 0.0125068 0.132019i 0.000637407 0.00672829i
\(386\) 0 0
\(387\) −5.43676 5.43676i −0.276366 0.276366i
\(388\) 0 0
\(389\) −23.8149 −1.20747 −0.603733 0.797187i \(-0.706321\pi\)
−0.603733 + 0.797187i \(0.706321\pi\)
\(390\) 0 0
\(391\) 8.32793 18.0705i 0.421161 0.913864i
\(392\) 0 0
\(393\) −0.829140 0.829140i −0.0418246 0.0418246i
\(394\) 0 0
\(395\) −9.88850 + 8.17705i −0.497544 + 0.411432i
\(396\) 0 0
\(397\) −3.99927 + 3.99927i −0.200718 + 0.200718i −0.800308 0.599590i \(-0.795330\pi\)
0.599590 + 0.800308i \(0.295330\pi\)
\(398\) 0 0
\(399\) 13.5312i 0.677409i
\(400\) 0 0
\(401\) 17.2957i 0.863704i −0.901944 0.431852i \(-0.857860\pi\)
0.901944 0.431852i \(-0.142140\pi\)
\(402\) 0 0
\(403\) −37.3233 37.3233i −1.85921 1.85921i
\(404\) 0 0
\(405\) 1.72321 1.42497i 0.0856272 0.0708073i
\(406\) 0 0
\(407\) −0.162781 + 0.162781i −0.00806876 + 0.00806876i
\(408\) 0 0
\(409\) 13.5440i 0.669708i −0.942270 0.334854i \(-0.891313\pi\)
0.942270 0.334854i \(-0.108687\pi\)
\(410\) 0 0
\(411\) 22.7731i 1.12332i
\(412\) 0 0
\(413\) 3.50544 3.50544i 0.172492 0.172492i
\(414\) 0 0
\(415\) −0.0544806 + 0.575082i −0.00267435 + 0.0282296i
\(416\) 0 0
\(417\) −13.2229 + 13.2229i −0.647530 + 0.647530i
\(418\) 0 0
\(419\) 36.4513 1.78076 0.890382 0.455215i \(-0.150438\pi\)
0.890382 + 0.455215i \(0.150438\pi\)
\(420\) 0 0
\(421\) 13.1308i 0.639958i −0.947425 0.319979i \(-0.896324\pi\)
0.947425 0.319979i \(-0.103676\pi\)
\(422\) 0 0
\(423\) −4.36786 4.36786i −0.212372 0.212372i
\(424\) 0 0
\(425\) 17.1619 11.6529i 0.832476 0.565249i
\(426\) 0 0
\(427\) 8.03231 8.03231i 0.388711 0.388711i
\(428\) 0 0
\(429\) 0.153431 0.00740772
\(430\) 0 0
\(431\) 16.3040i 0.785337i −0.919680 0.392668i \(-0.871552\pi\)
0.919680 0.392668i \(-0.128448\pi\)
\(432\) 0 0
\(433\) 6.99454 6.99454i 0.336136 0.336136i −0.518775 0.854911i \(-0.673612\pi\)
0.854911 + 0.518775i \(0.173612\pi\)
\(434\) 0 0
\(435\) 1.52217 16.0676i 0.0729826 0.770383i
\(436\) 0 0
\(437\) 9.52114 + 25.7985i 0.455458 + 1.23411i
\(438\) 0 0
\(439\) 24.8220i 1.18469i 0.805684 + 0.592345i \(0.201798\pi\)
−0.805684 + 0.592345i \(0.798202\pi\)
\(440\) 0 0
\(441\) 1.43123 0.0681538
\(442\) 0 0
\(443\) −19.6606 19.6606i −0.934102 0.934102i 0.0638575 0.997959i \(-0.479660\pi\)
−0.997959 + 0.0638575i \(0.979660\pi\)
\(444\) 0 0
\(445\) −17.7310 + 14.6622i −0.840532 + 0.695057i
\(446\) 0 0
\(447\) −2.77247 2.77247i −0.131133 0.131133i
\(448\) 0 0
\(449\) 5.35828i 0.252873i −0.991975 0.126436i \(-0.959646\pi\)
0.991975 0.126436i \(-0.0403540\pi\)
\(450\) 0 0
\(451\) 0.00155984i 7.34498e-5i
\(452\) 0 0
\(453\) 4.51294 + 4.51294i 0.212036 + 0.212036i
\(454\) 0 0
\(455\) 20.5299 + 24.8268i 0.962459 + 1.16390i
\(456\) 0 0
\(457\) 11.7631 + 11.7631i 0.550255 + 0.550255i 0.926514 0.376259i \(-0.122790\pi\)
−0.376259 + 0.926514i \(0.622790\pi\)
\(458\) 0 0
\(459\) −4.14884 −0.193651
\(460\) 0 0
\(461\) 28.6561 1.33465 0.667325 0.744767i \(-0.267439\pi\)
0.667325 + 0.744767i \(0.267439\pi\)
\(462\) 0 0
\(463\) −22.3062 22.3062i −1.03666 1.03666i −0.999302 0.0373556i \(-0.988107\pi\)
−0.0373556 0.999302i \(-0.511893\pi\)
\(464\) 0 0
\(465\) 19.2458 + 1.82326i 0.892505 + 0.0845519i
\(466\) 0 0
\(467\) 8.67121 + 8.67121i 0.401256 + 0.401256i 0.878675 0.477420i \(-0.158428\pi\)
−0.477420 + 0.878675i \(0.658428\pi\)
\(468\) 0 0
\(469\) 0.493511i 0.0227882i
\(470\) 0 0
\(471\) 14.2274i 0.655566i
\(472\) 0 0
\(473\) 0.136631 + 0.136631i 0.00628231 + 0.00628231i
\(474\) 0 0
\(475\) −5.38382 + 28.1600i −0.247027 + 1.29207i
\(476\) 0 0
\(477\) −9.27165 9.27165i −0.424520 0.424520i
\(478\) 0 0
\(479\) −14.9417 −0.682703 −0.341352 0.939936i \(-0.610885\pi\)
−0.341352 + 0.939936i \(0.610885\pi\)
\(480\) 0 0
\(481\) 55.9257i 2.54999i
\(482\) 0 0
\(483\) −10.6173 + 3.91841i −0.483105 + 0.178294i
\(484\) 0 0
\(485\) 40.5726 + 3.84366i 1.84230 + 0.174532i
\(486\) 0 0
\(487\) 2.71591 2.71591i 0.123070 0.123070i −0.642889 0.765959i \(-0.722265\pi\)
0.765959 + 0.642889i \(0.222265\pi\)
\(488\) 0 0
\(489\) 22.0017i 0.994950i
\(490\) 0 0
\(491\) 10.8186 0.488238 0.244119 0.969745i \(-0.421501\pi\)
0.244119 + 0.969745i \(0.421501\pi\)
\(492\) 0 0
\(493\) −21.1748 + 21.1748i −0.953663 + 0.953663i
\(494\) 0 0
\(495\) −0.0433061 + 0.0358109i −0.00194647 + 0.00160958i
\(496\) 0 0
\(497\) 7.44091 + 7.44091i 0.333770 + 0.333770i
\(498\) 0 0
\(499\) 8.18810i 0.366550i 0.983062 + 0.183275i \(0.0586698\pi\)
−0.983062 + 0.183275i \(0.941330\pi\)
\(500\) 0 0
\(501\) 17.7947 0.795008
\(502\) 0 0
\(503\) −6.69930 + 6.69930i −0.298707 + 0.298707i −0.840507 0.541800i \(-0.817743\pi\)
0.541800 + 0.840507i \(0.317743\pi\)
\(504\) 0 0
\(505\) −27.0924 + 22.4034i −1.20560 + 0.996939i
\(506\) 0 0
\(507\) −17.1643 + 17.1643i −0.762293 + 0.762293i
\(508\) 0 0
\(509\) 11.2505i 0.498672i −0.968417 0.249336i \(-0.919788\pi\)
0.968417 0.249336i \(-0.0802123\pi\)
\(510\) 0 0
\(511\) 5.28200i 0.233662i
\(512\) 0 0
\(513\) 4.05455 4.05455i 0.179013 0.179013i
\(514\) 0 0
\(515\) 0.833593 8.79917i 0.0367325 0.387738i
\(516\) 0 0
\(517\) 0.109769 + 0.109769i 0.00482762 + 0.00482762i
\(518\) 0 0
\(519\) 7.74389i 0.339919i
\(520\) 0 0
\(521\) 28.5201i 1.24949i 0.780829 + 0.624745i \(0.214797\pi\)
−0.780829 + 0.624745i \(0.785203\pi\)
\(522\) 0 0
\(523\) −24.3263 + 24.3263i −1.06371 + 1.06371i −0.0658877 + 0.997827i \(0.520988\pi\)
−0.997827 + 0.0658877i \(0.979012\pi\)
\(524\) 0 0
\(525\) −11.5892 2.21571i −0.505795 0.0967013i
\(526\) 0 0
\(527\) −25.3632 25.3632i −1.10484 1.10484i
\(528\) 0 0
\(529\) −17.4857 + 14.9416i −0.760247 + 0.649634i
\(530\) 0 0
\(531\) −2.10077 −0.0911656
\(532\) 0 0
\(533\) −0.267952 0.267952i −0.0116063 0.0116063i
\(534\) 0 0
\(535\) −0.00903602 0.000856032i −0.000390661 3.70095e-5i
\(536\) 0 0
\(537\) −3.81830 + 3.81830i −0.164772 + 0.164772i
\(538\) 0 0
\(539\) −0.0359683 −0.00154926
\(540\) 0 0
\(541\) 18.2453 0.784426 0.392213 0.919874i \(-0.371710\pi\)
0.392213 + 0.919874i \(0.371710\pi\)
\(542\) 0 0
\(543\) 2.72329 2.72329i 0.116868 0.116868i
\(544\) 0 0
\(545\) 8.33453 6.89203i 0.357012 0.295222i
\(546\) 0 0
\(547\) 7.56332 7.56332i 0.323384 0.323384i −0.526680 0.850064i \(-0.676563\pi\)
0.850064 + 0.526680i \(0.176563\pi\)
\(548\) 0 0
\(549\) −4.81366 −0.205442
\(550\) 0 0
\(551\) 41.3870i 1.76315i
\(552\) 0 0
\(553\) −9.57538 9.57538i −0.407187 0.407187i
\(554\) 0 0
\(555\) 13.0531 + 15.7851i 0.554074 + 0.670041i
\(556\) 0 0
\(557\) −8.46308 8.46308i −0.358592 0.358592i 0.504702 0.863294i \(-0.331602\pi\)
−0.863294 + 0.504702i \(0.831602\pi\)
\(558\) 0 0
\(559\) −46.9416 −1.98542
\(560\) 0 0
\(561\) 0.104265 0.00440206
\(562\) 0 0
\(563\) 6.86406 6.86406i 0.289286 0.289286i −0.547512 0.836798i \(-0.684425\pi\)
0.836798 + 0.547512i \(0.184425\pi\)
\(564\) 0 0
\(565\) −1.13034 + 11.9315i −0.0475538 + 0.501964i
\(566\) 0 0
\(567\) 1.66865 + 1.66865i 0.0700766 + 0.0700766i
\(568\) 0 0
\(569\) −29.0267 −1.21686 −0.608431 0.793607i \(-0.708201\pi\)
−0.608431 + 0.793607i \(0.708201\pi\)
\(570\) 0 0
\(571\) 34.8226i 1.45728i −0.684896 0.728641i \(-0.740152\pi\)
0.684896 0.728641i \(-0.259848\pi\)
\(572\) 0 0
\(573\) −1.78319 + 1.78319i −0.0744939 + 0.0744939i
\(574\) 0 0
\(575\) −23.6549 + 3.93022i −0.986477 + 0.163901i
\(576\) 0 0
\(577\) −5.76114 + 5.76114i −0.239840 + 0.239840i −0.816784 0.576944i \(-0.804245\pi\)
0.576944 + 0.816784i \(0.304245\pi\)
\(578\) 0 0
\(579\) 23.2541i 0.966408i
\(580\) 0 0
\(581\) −0.609627 −0.0252916
\(582\) 0 0
\(583\) 0.233006 + 0.233006i 0.00965013 + 0.00965013i
\(584\) 0 0
\(585\) 1.28754 13.5909i 0.0532332 0.561914i
\(586\) 0 0
\(587\) −27.3927 + 27.3927i −1.13062 + 1.13062i −0.140542 + 0.990075i \(0.544885\pi\)
−0.990075 + 0.140542i \(0.955115\pi\)
\(588\) 0 0
\(589\) 49.5736 2.04264
\(590\) 0 0
\(591\) −15.6398 −0.643337
\(592\) 0 0
\(593\) −0.308745 0.308745i −0.0126786 0.0126786i 0.700739 0.713418i \(-0.252854\pi\)
−0.713418 + 0.700739i \(0.752854\pi\)
\(594\) 0 0
\(595\) 13.9512 + 16.8712i 0.571944 + 0.691651i
\(596\) 0 0
\(597\) 8.86990 + 8.86990i 0.363021 + 0.363021i
\(598\) 0 0
\(599\) 34.7873i 1.42137i 0.703510 + 0.710686i \(0.251615\pi\)
−0.703510 + 0.710686i \(0.748385\pi\)
\(600\) 0 0
\(601\) −13.9756 −0.570076 −0.285038 0.958516i \(-0.592006\pi\)
−0.285038 + 0.958516i \(0.592006\pi\)
\(602\) 0 0
\(603\) −0.147877 + 0.147877i −0.00602204 + 0.00602204i
\(604\) 0 0
\(605\) −18.9543 + 15.6738i −0.770600 + 0.637229i
\(606\) 0 0
\(607\) 10.8569 10.8569i 0.440667 0.440667i −0.451569 0.892236i \(-0.649136\pi\)
0.892236 + 0.451569i \(0.149136\pi\)
\(608\) 0 0
\(609\) 17.0328 0.690204
\(610\) 0 0
\(611\) −37.7126 −1.52569
\(612\) 0 0
\(613\) −19.7883 + 19.7883i −0.799244 + 0.799244i −0.982976 0.183733i \(-0.941182\pi\)
0.183733 + 0.982976i \(0.441182\pi\)
\(614\) 0 0
\(615\) 0.138170 + 0.0130896i 0.00557155 + 0.000527823i
\(616\) 0 0
\(617\) −17.8650 17.8650i −0.719218 0.719218i 0.249227 0.968445i \(-0.419823\pi\)
−0.968445 + 0.249227i \(0.919823\pi\)
\(618\) 0 0
\(619\) −20.9569 −0.842330 −0.421165 0.906984i \(-0.638379\pi\)
−0.421165 + 0.906984i \(0.638379\pi\)
\(620\) 0 0
\(621\) 4.35555 + 2.00729i 0.174782 + 0.0805497i
\(622\) 0 0
\(623\) −17.1696 17.1696i −0.687885 0.687885i
\(624\) 0 0
\(625\) −23.2368 9.22225i −0.929473 0.368890i
\(626\) 0 0
\(627\) −0.101895 + 0.101895i −0.00406930 + 0.00406930i
\(628\) 0 0
\(629\) 38.0046i 1.51534i
\(630\) 0 0
\(631\) 43.1937i 1.71952i 0.510702 + 0.859758i \(0.329386\pi\)
−0.510702 + 0.859758i \(0.670614\pi\)
\(632\) 0 0
\(633\) 6.31147 + 6.31147i 0.250859 + 0.250859i
\(634\) 0 0
\(635\) 1.50051 15.8389i 0.0595459 0.628549i
\(636\) 0 0
\(637\) 6.17870 6.17870i 0.244809 0.244809i
\(638\) 0 0
\(639\) 4.45924i 0.176405i
\(640\) 0 0
\(641\) 3.13450i 0.123805i 0.998082 + 0.0619027i \(0.0197169\pi\)
−0.998082 + 0.0619027i \(0.980283\pi\)
\(642\) 0 0
\(643\) 24.2196 24.2196i 0.955128 0.955128i −0.0439078 0.999036i \(-0.513981\pi\)
0.999036 + 0.0439078i \(0.0139808\pi\)
\(644\) 0 0
\(645\) 13.2493 10.9562i 0.521692 0.431400i
\(646\) 0 0
\(647\) 13.2492 13.2492i 0.520879 0.520879i −0.396958 0.917837i \(-0.629934\pi\)
0.917837 + 0.396958i \(0.129934\pi\)
\(648\) 0 0
\(649\) 0.0527945 0.00207237
\(650\) 0 0
\(651\) 20.4020i 0.799616i
\(652\) 0 0
\(653\) −3.89395 3.89395i −0.152382 0.152382i 0.626799 0.779181i \(-0.284365\pi\)
−0.779181 + 0.626799i \(0.784365\pi\)
\(654\) 0 0
\(655\) 2.02061 1.67089i 0.0789517 0.0652872i
\(656\) 0 0
\(657\) 1.58272 1.58272i 0.0617477 0.0617477i
\(658\) 0 0
\(659\) −3.97910 −0.155004 −0.0775019 0.996992i \(-0.524694\pi\)
−0.0775019 + 0.996992i \(0.524694\pi\)
\(660\) 0 0
\(661\) 15.3507i 0.597075i −0.954398 0.298537i \(-0.903501\pi\)
0.954398 0.298537i \(-0.0964988\pi\)
\(662\) 0 0
\(663\) −17.9108 + 17.9108i −0.695598 + 0.695598i
\(664\) 0 0
\(665\) −30.1219 2.85361i −1.16808 0.110658i
\(666\) 0 0
\(667\) 32.4745 11.9850i 1.25742 0.464060i
\(668\) 0 0
\(669\) 11.4500i 0.442682i
\(670\) 0 0
\(671\) 0.120972 0.00467009
\(672\) 0 0
\(673\) −0.494649 0.494649i −0.0190673 0.0190673i 0.697509 0.716576i \(-0.254292\pi\)
−0.716576 + 0.697509i \(0.754292\pi\)
\(674\) 0 0
\(675\) 2.80871 + 4.13656i 0.108107 + 0.159216i
\(676\) 0 0
\(677\) −21.8101 21.8101i −0.838230 0.838230i 0.150396 0.988626i \(-0.451945\pi\)
−0.988626 + 0.150396i \(0.951945\pi\)
\(678\) 0 0
\(679\) 43.0098i 1.65056i
\(680\) 0 0
\(681\) 27.1412i 1.04005i
\(682\) 0 0
\(683\) 15.1043 + 15.1043i 0.577948 + 0.577948i 0.934338 0.356389i \(-0.115992\pi\)
−0.356389 + 0.934338i \(0.615992\pi\)
\(684\) 0 0
\(685\) −50.6953 4.80264i −1.93697 0.183500i
\(686\) 0 0
\(687\) 12.5290 + 12.5290i 0.478011 + 0.478011i
\(688\) 0 0
\(689\) −80.0525 −3.04976
\(690\) 0 0
\(691\) 28.5555 1.08630 0.543151 0.839635i \(-0.317231\pi\)
0.543151 + 0.839635i \(0.317231\pi\)
\(692\) 0 0
\(693\) −0.0419348 0.0419348i −0.00159297 0.00159297i
\(694\) 0 0
\(695\) −26.6470 32.2242i −1.01078 1.22233i
\(696\) 0 0
\(697\) −0.182088 0.182088i −0.00689706 0.00689706i
\(698\) 0 0
\(699\) 3.01089i 0.113882i
\(700\) 0 0
\(701\) 5.24702i 0.198177i 0.995079 + 0.0990887i \(0.0315928\pi\)
−0.995079 + 0.0990887i \(0.968407\pi\)
\(702\) 0 0
\(703\) 37.1408 + 37.1408i 1.40079 + 1.40079i
\(704\) 0 0
\(705\) 10.6444 8.80215i 0.400892 0.331508i
\(706\) 0 0
\(707\) −26.2346 26.2346i −0.986652 0.986652i
\(708\) 0 0
\(709\) −1.96518 −0.0738038 −0.0369019 0.999319i \(-0.511749\pi\)
−0.0369019 + 0.999319i \(0.511749\pi\)
\(710\) 0 0
\(711\) 5.73841i 0.215207i
\(712\) 0 0
\(713\) 14.3556 + 38.8981i 0.537623 + 1.45674i
\(714\) 0 0
\(715\) −0.0323572 + 0.341553i −0.00121009 + 0.0127734i
\(716\) 0 0
\(717\) −17.3410 + 17.3410i −0.647613 + 0.647613i
\(718\) 0 0
\(719\) 30.7947i 1.14845i 0.818698 + 0.574224i \(0.194696\pi\)
−0.818698 + 0.574224i \(0.805304\pi\)
\(720\) 0 0
\(721\) 9.32774 0.347383
\(722\) 0 0
\(723\) −13.2090 + 13.2090i −0.491249 + 0.491249i
\(724\) 0 0
\(725\) 35.4471 + 6.77702i 1.31647 + 0.251692i
\(726\) 0 0
\(727\) 5.78767 + 5.78767i 0.214653 + 0.214653i 0.806241 0.591588i \(-0.201499\pi\)
−0.591588 + 0.806241i \(0.701499\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −31.8993 −1.17984
\(732\) 0 0
\(733\) 23.2686 23.2686i 0.859445 0.859445i −0.131828 0.991273i \(-0.542085\pi\)
0.991273 + 0.131828i \(0.0420845\pi\)
\(734\) 0 0
\(735\) −0.301833 + 3.18606i −0.0111333 + 0.117520i
\(736\) 0 0
\(737\) 0.00371631 0.00371631i 0.000136892 0.000136892i
\(738\) 0 0
\(739\) 38.1921i 1.40492i 0.711725 + 0.702459i \(0.247914\pi\)
−0.711725 + 0.702459i \(0.752086\pi\)
\(740\) 0 0
\(741\) 35.0075i 1.28603i
\(742\) 0 0
\(743\) 6.32639 6.32639i 0.232093 0.232093i −0.581473 0.813566i \(-0.697523\pi\)
0.813566 + 0.581473i \(0.197523\pi\)
\(744\) 0 0
\(745\) 6.75649 5.58711i 0.247539 0.204696i
\(746\) 0 0
\(747\) 0.182671 + 0.182671i 0.00668358 + 0.00668358i
\(748\) 0 0
\(749\) 0.00957882i 0.000350002i
\(750\) 0 0
\(751\) 43.1418i 1.57427i 0.616783 + 0.787134i \(0.288436\pi\)
−0.616783 + 0.787134i \(0.711564\pi\)
\(752\) 0 0
\(753\) 1.85182 1.85182i 0.0674842 0.0674842i
\(754\) 0 0
\(755\) −10.9980 + 9.09452i −0.400258 + 0.330984i
\(756\) 0 0
\(757\) −24.0750 24.0750i −0.875021 0.875021i 0.117993 0.993014i \(-0.462354\pi\)
−0.993014 + 0.117993i \(0.962354\pi\)
\(758\) 0 0
\(759\) −0.109459 0.0504453i −0.00397312 0.00183105i
\(760\) 0 0
\(761\) −38.3892 −1.39161 −0.695804 0.718232i \(-0.744952\pi\)
−0.695804 + 0.718232i \(0.744952\pi\)
\(762\) 0 0
\(763\) 8.07061 + 8.07061i 0.292176 + 0.292176i
\(764\) 0 0
\(765\) 0.874953 9.23574i 0.0316340 0.333919i
\(766\) 0 0
\(767\) −9.06914 + 9.06914i −0.327468 + 0.327468i
\(768\) 0 0
\(769\) 25.0561 0.903547 0.451773 0.892133i \(-0.350792\pi\)
0.451773 + 0.892133i \(0.350792\pi\)
\(770\) 0 0
\(771\) 29.5969 1.06591
\(772\) 0 0
\(773\) −30.8282 + 30.8282i −1.10881 + 1.10881i −0.115505 + 0.993307i \(0.536849\pi\)
−0.993307 + 0.115505i \(0.963151\pi\)
\(774\) 0 0
\(775\) −8.11754 + 42.4587i −0.291591 + 1.52516i
\(776\) 0 0
\(777\) −15.2853 + 15.2853i −0.548356 + 0.548356i
\(778\) 0 0
\(779\) 0.355899 0.0127514
\(780\) 0 0
\(781\) 0.112065i 0.00401002i
\(782\) 0 0
\(783\) −5.10377 5.10377i −0.182394 0.182394i
\(784\) 0 0
\(785\) −31.6717 3.00044i −1.13041 0.107090i
\(786\) 0 0
\(787\) −9.15839 9.15839i −0.326461 0.326461i 0.524778 0.851239i \(-0.324148\pi\)
−0.851239 + 0.524778i \(0.824148\pi\)
\(788\) 0 0
\(789\) −7.41395 −0.263944
\(790\) 0 0
\(791\) −12.6483 −0.449721
\(792\) 0 0
\(793\) −20.7809 + 20.7809i −0.737950 + 0.737950i
\(794\) 0 0
\(795\) 22.5949 18.6843i 0.801360 0.662665i
\(796\) 0 0
\(797\) 5.74270 + 5.74270i 0.203417 + 0.203417i 0.801462 0.598045i \(-0.204056\pi\)
−0.598045 + 0.801462i \(0.704056\pi\)
\(798\) 0 0
\(799\) −25.6277 −0.906644
\(800\) 0 0
\(801\) 10.2895i 0.363562i
\(802\) 0 0
\(803\) −0.0397753 + 0.0397753i −0.00140364 + 0.00140364i
\(804\) 0 0
\(805\) −6.48369 24.4616i −0.228520 0.862158i
\(806\) 0 0
\(807\) 11.6458 11.6458i 0.409953 0.409953i
\(808\) 0 0
\(809\) 38.6204i 1.35782i 0.734222 + 0.678910i \(0.237547\pi\)
−0.734222 + 0.678910i \(0.762453\pi\)
\(810\) 0 0
\(811\) 41.5982 1.46071 0.730356 0.683067i \(-0.239354\pi\)
0.730356 + 0.683067i \(0.239354\pi\)
\(812\) 0 0
\(813\) −11.8511 11.8511i −0.415637 0.415637i
\(814\) 0 0
\(815\) 48.9779 + 4.63995i 1.71562 + 0.162530i
\(816\) 0 0
\(817\) 31.1744 31.1744i 1.09065 1.09065i
\(818\) 0 0
\(819\) 14.4073 0.503432
\(820\) 0 0
\(821\) 19.2961 0.673440 0.336720 0.941605i \(-0.390682\pi\)
0.336720 + 0.941605i \(0.390682\pi\)
\(822\) 0 0
\(823\) −0.457979 0.457979i −0.0159642 0.0159642i 0.699080 0.715044i \(-0.253593\pi\)
−0.715044 + 0.699080i \(0.753593\pi\)
\(824\) 0 0
\(825\) −0.0705859 0.103956i −0.00245749 0.00361928i
\(826\) 0 0
\(827\) 7.39649 + 7.39649i 0.257201 + 0.257201i 0.823915 0.566714i \(-0.191785\pi\)
−0.566714 + 0.823915i \(0.691785\pi\)
\(828\) 0 0
\(829\) 35.0850i 1.21855i 0.792958 + 0.609276i \(0.208540\pi\)
−0.792958 + 0.609276i \(0.791460\pi\)
\(830\) 0 0
\(831\) −21.0452 −0.730050
\(832\) 0 0
\(833\) 4.19876 4.19876i 0.145479 0.145479i
\(834\) 0 0
\(835\) −3.75273 + 39.6127i −0.129869 + 1.37086i
\(836\) 0 0
\(837\) 6.11332 6.11332i 0.211307 0.211307i
\(838\) 0 0
\(839\) −12.1734 −0.420274 −0.210137 0.977672i \(-0.567391\pi\)
−0.210137 + 0.977672i \(0.567391\pi\)
\(840\) 0 0
\(841\) −23.0970 −0.796448
\(842\) 0 0
\(843\) −2.16902 + 2.16902i −0.0747050 + 0.0747050i
\(844\) 0 0
\(845\) −34.5897 41.8292i −1.18992 1.43897i
\(846\) 0 0
\(847\) −18.3541 18.3541i −0.630653 0.630653i
\(848\) 0 0
\(849\) 9.34256 0.320636
\(850\) 0 0
\(851\) −18.3873 + 39.8980i −0.630309 + 1.36769i
\(852\) 0 0
\(853\) 28.7887 + 28.7887i 0.985708 + 0.985708i 0.999899 0.0141916i \(-0.00451747\pi\)
−0.0141916 + 0.999899i \(0.504517\pi\)
\(854\) 0 0
\(855\) 8.17078 + 9.88091i 0.279435 + 0.337920i
\(856\) 0 0
\(857\) −31.7818 + 31.7818i −1.08564 + 1.08564i −0.0896730 + 0.995971i \(0.528582\pi\)
−0.995971 + 0.0896730i \(0.971418\pi\)
\(858\) 0 0
\(859\) 14.5207i 0.495441i −0.968832 0.247721i \(-0.920319\pi\)
0.968832 0.247721i \(-0.0796815\pi\)
\(860\) 0 0
\(861\) 0.146470i 0.00499168i
\(862\) 0 0
\(863\) 34.7792 + 34.7792i 1.18390 + 1.18390i 0.978726 + 0.205170i \(0.0657748\pi\)
0.205170 + 0.978726i \(0.434225\pi\)
\(864\) 0 0
\(865\) 17.2387 + 1.63311i 0.586132 + 0.0555275i
\(866\) 0 0
\(867\) −0.150544 + 0.150544i −0.00511273 + 0.00511273i
\(868\) 0 0
\(869\) 0.144212i 0.00489206i
\(870\) 0 0
\(871\) 1.27679i 0.0432624i
\(872\) 0 0
\(873\) 12.8876 12.8876i 0.436180 0.436180i
\(874\) 0 0
\(875\) 7.37644 25.3315i 0.249369 0.856361i
\(876\) 0 0
\(877\) −33.7322 + 33.7322i −1.13906 + 1.13906i −0.150435 + 0.988620i \(0.548067\pi\)
−0.988620 + 0.150435i \(0.951933\pi\)
\(878\) 0 0
\(879\) −30.2595 −1.02063
\(880\) 0 0
\(881\) 42.1641i 1.42054i −0.703927 0.710272i \(-0.748572\pi\)
0.703927 0.710272i \(-0.251428\pi\)
\(882\) 0 0
\(883\) −13.2200 13.2200i −0.444887 0.444887i 0.448763 0.893651i \(-0.351865\pi\)
−0.893651 + 0.448763i \(0.851865\pi\)
\(884\) 0 0
\(885\) 0.443033 4.67652i 0.0148924 0.157200i
\(886\) 0 0
\(887\) −18.1329 + 18.1329i −0.608845 + 0.608845i −0.942644 0.333799i \(-0.891669\pi\)
0.333799 + 0.942644i \(0.391669\pi\)
\(888\) 0 0
\(889\) 16.7904 0.563132
\(890\) 0 0
\(891\) 0.0251310i 0.000841921i
\(892\) 0 0
\(893\) 25.0453 25.0453i 0.838109 0.838109i
\(894\) 0 0
\(895\) −7.69468 9.30517i −0.257205 0.311038i
\(896\) 0 0
\(897\) 27.4687 10.1376i 0.917154 0.338483i
\(898\) 0 0
\(899\) 62.4020i 2.08122i
\(900\) 0 0
\(901\) −54.4000 −1.81233
\(902\) 0 0
\(903\) 12.8298 + 12.8298i 0.426949 + 0.426949i
\(904\) 0 0
\(905\) 5.48800 + 6.63663i 0.182427 + 0.220609i
\(906\) 0 0
\(907\) −7.94315 7.94315i −0.263748 0.263748i 0.562827 0.826575i \(-0.309714\pi\)
−0.826575 + 0.562827i \(0.809714\pi\)
\(908\) 0 0
\(909\) 15.7220i 0.521467i
\(910\) 0 0
\(911\) 6.75307i 0.223739i 0.993723 + 0.111870i \(0.0356839\pi\)
−0.993723 + 0.111870i \(0.964316\pi\)
\(912\) 0 0
\(913\) −0.00459071 0.00459071i −0.000151930 0.000151930i
\(914\) 0 0
\(915\) 1.01516 10.7157i 0.0335601 0.354250i
\(916\) 0 0
\(917\) 1.95663 + 1.95663i 0.0646135 + 0.0646135i
\(918\) 0 0
\(919\) 25.5597 0.843138 0.421569 0.906796i \(-0.361480\pi\)
0.421569 + 0.906796i \(0.361480\pi\)
\(920\) 0 0
\(921\) 13.3370 0.439470
\(922\) 0 0
\(923\) −19.2508 19.2508i −0.633648 0.633648i
\(924\) 0 0
\(925\) −37.8920 + 25.7286i −1.24588 + 0.845951i
\(926\) 0 0
\(927\) −2.79500 2.79500i −0.0917998 0.0917998i
\(928\) 0 0
\(929\) 13.2581i 0.434985i 0.976062 + 0.217492i \(0.0697877\pi\)
−0.976062 + 0.217492i \(0.930212\pi\)
\(930\) 0 0
\(931\) 8.20668i 0.268963i
\(932\) 0 0
\(933\) −12.7752 12.7752i −0.418240 0.418240i
\(934\) 0 0
\(935\) −0.0219885 + 0.232104i −0.000719100 + 0.00759061i
\(936\) 0 0
\(937\) 10.7271 + 10.7271i 0.350441 + 0.350441i 0.860273 0.509833i \(-0.170293\pi\)
−0.509833 + 0.860273i \(0.670293\pi\)
\(938\) 0 0
\(939\) 1.59162 0.0519405
\(940\) 0 0
\(941\) 45.1199i 1.47087i −0.677597 0.735434i \(-0.736979\pi\)
0.677597 0.735434i \(-0.263021\pi\)
\(942\) 0 0
\(943\) 0.103062 + 0.279257i 0.00335617 + 0.00909386i
\(944\) 0 0
\(945\) −4.06648 + 3.36268i −0.132283 + 0.109388i
\(946\) 0 0
\(947\) 19.0586 19.0586i 0.619320 0.619320i −0.326037 0.945357i \(-0.605713\pi\)
0.945357 + 0.326037i \(0.105713\pi\)
\(948\) 0 0
\(949\) 13.6654i 0.443597i
\(950\) 0 0
\(951\) −12.6019 −0.408644
\(952\) 0 0
\(953\) −8.48535 + 8.48535i −0.274868 + 0.274868i −0.831056 0.556189i \(-0.812263\pi\)
0.556189 + 0.831056i \(0.312263\pi\)
\(954\) 0 0
\(955\) −3.59350 4.34562i −0.116283 0.140621i
\(956\) 0 0
\(957\) 0.128263 + 0.128263i 0.00414616 + 0.00414616i
\(958\) 0 0
\(959\) 53.7406i 1.73537i
\(960\) 0 0
\(961\) 43.7453 1.41114
\(962\) 0 0
\(963\) −0.00287023 + 0.00287023i −9.24920e−5 + 9.24920e-5i
\(964\) 0 0
\(965\) 51.7660 + 4.90408i 1.66641 + 0.157868i
\(966\) 0 0
\(967\) 35.8055 35.8055i 1.15143 1.15143i 0.165160 0.986267i \(-0.447186\pi\)
0.986267 0.165160i \(-0.0528140\pi\)
\(968\) 0 0
\(969\) 23.7895i 0.764229i
\(970\) 0 0
\(971\) 58.3681i 1.87312i −0.350506 0.936561i \(-0.613990\pi\)
0.350506 0.936561i \(-0.386010\pi\)
\(972\) 0 0
\(973\) 31.2038 31.2038i 1.00035 1.00035i
\(974\) 0 0
\(975\) 29.9831 + 5.73238i 0.960229 + 0.183583i
\(976\) 0 0
\(977\) −40.7797 40.7797i −1.30466 1.30466i −0.925216 0.379441i \(-0.876116\pi\)
−0.379441 0.925216i \(-0.623884\pi\)
\(978\) 0 0
\(979\) 0.258586i 0.00826445i
\(980\) 0 0
\(981\) 4.83662i 0.154421i
\(982\) 0 0
\(983\) 25.9294 25.9294i 0.827019 0.827019i −0.160085 0.987103i \(-0.551177\pi\)
0.987103 + 0.160085i \(0.0511767\pi\)
\(984\) 0 0
\(985\) 3.29830 34.8159i 0.105092 1.10933i
\(986\) 0 0
\(987\) 10.3074 + 10.3074i 0.328087 + 0.328087i
\(988\) 0 0
\(989\) 33.4886 + 15.4335i 1.06488 + 0.490757i
\(990\) 0 0
\(991\) 21.6862 0.688885 0.344443 0.938807i \(-0.388068\pi\)
0.344443 + 0.938807i \(0.388068\pi\)
\(992\) 0 0
\(993\) 12.7480 + 12.7480i 0.404545 + 0.404545i
\(994\) 0 0
\(995\) −21.6159 + 17.8747i −0.685269 + 0.566666i
\(996\) 0 0
\(997\) 20.7469 20.7469i 0.657059 0.657059i −0.297624 0.954683i \(-0.596194\pi\)
0.954683 + 0.297624i \(0.0961941\pi\)
\(998\) 0 0
\(999\) 9.16028 0.289818
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1380.2.t.a.1333.23 yes 48
5.2 odd 4 inner 1380.2.t.a.1057.24 yes 48
23.22 odd 2 inner 1380.2.t.a.1333.24 yes 48
115.22 even 4 inner 1380.2.t.a.1057.23 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.t.a.1057.23 48 115.22 even 4 inner
1380.2.t.a.1057.24 yes 48 5.2 odd 4 inner
1380.2.t.a.1333.23 yes 48 1.1 even 1 trivial
1380.2.t.a.1333.24 yes 48 23.22 odd 2 inner